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Linear Algebra and its Applications 432 (2010) 546–555

Contents lists available at ScienceDirect

Linear Algebra and its Applications

j ourna l homepage: www.e lsev ie r .com/ loca te / laa

On elementary divisors perturbation of nonnegative

matrices�

Javier Ccapa, Ricardo L. Soto∗

Dpto. Matemáticas, Universidad Católica del Norte, Casilla, 1280, Antofagasta, Chile

A R T I C L E I N F O A B S T R A C T

Article history:

Received 23 March 2009

Accepted 1 September 2009

Available online 1 October 2009

Submitted by R.A. Brualdi

AMS classification:

15A18

Keywords:

Nonnegative matrices

Elementary divisors

Spectra perturbation

An outstanding result of Guo (1997) [3] establishes that if the list

Λ = {λ1, λ2, . . . , λn} is the spectrum of an n × n nonnegative ma-

trix, where λ1 is its Perron eigenvalue and λ2 ∈ R, then for any

t � 0, the list Λt = {λ1 + t, λ2 ± t, . . . , λn} is also the spectrum of

a nonnegative matrix. In this paper we extend the result of Guo

to elementary divisors. In particular, if A is a nonnegative matrix

with spectrum Λ then, by means of two rank one perturbations,

we construct a modified matrix B, which is also nonnegative, with

spectrum Λt and we explicitly provide the Jordan canonical form

of B.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Let A ∈ Cn×n and let

J(A) = S−1AS =

⎡⎢⎢⎢⎣

Jn1(λ1)

Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎦

be the Jordan canonical form of A (hereafter JCF of A). The ni × ni submatrices

�Supported by Fondecyt 1085125, Chile.

∗ Corresponding author. Fax: +56 55 355599.

E-mail addresses: jccapa@ucn.cl (J. Ccapa), rsoto@ucn.cl (R.L. Soto).

0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.laa.2009.09.001

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J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555 547

Jni(λi) =

⎡⎢⎢⎢⎢⎢⎣

λi 1

λi

. . .

. . . 1

λi

⎤⎥⎥⎥⎥⎥⎦, i = 1, 2, . . . , k

are called the Jordan blocks of J(A). Then the elementary divisors of A are the polynomials (λ − λi)ni ,

that is, the characteristic polynomials of Jni(λi), i = 1, . . . , k.A matrix A = (aij)

ni,j=1 is said to have constant row sums if all its rows add up to the same constant

α, i.e.∑n

j=1 aij = α, i = 1, . . . , n. The set of all matrices with constant row sums equal to α is denoted

by CSα . It is clear that any matrix A ∈ CSα has the eigenvector e = (1, 1, . . . , 1)T corresponding to

the eigenvalue α. Denote by ek the vector with 1 in the k-th position and zeros elsewhere, by σ(A)the spectrum of A, by ma(λ) the algebraic multiplicity of λ, and by Eij the matrix with 1 in position

(i, j) and zeros elsewhere. A nonnegative matrix A (aij � 0) is called stochastic if A ∈ CS1 and doubly

stochastic if A, AT ∈ CS1. For lack of simplicity we shall call nonnegative generalized stochastic to a

nonnegative matrix A ∈ CSα and nonnegative generalized doubly stochastic to a nonnegative matrix

Awith A, AT ∈ CSα .

In [4], Johnson proved that if Λ = {λ1, λ2, . . . , λn} is the spectrum of an n × n entrywise nonneg-

ative matrix, then Λ is also the spectrum of a nonnegative matrix with constant row sums equal to its

Perron eigenvalue. The following outstanding result of Brauer [1] has shown to be a very useful tool to

construct nonnegative matrices with prescribed spectrum.

Theorem 1.1 [1]. Let A be an n × n arbitrary matrix with eigenvalues λ1, λ2, . . . , λn. Let v = (v1, v2, . . . ,vn)

T be an eigenvector of A associated with the eigenvalue λk and let q be any n-dimensional vector. Thenthe matrix A + vqT has eigenvalues λ1, λ2, . . . , λk−1, λk + vTq, λk+1, . . . , λn.

In [3], Guo proved the following notable result:

Theorem1.2 [3]. If the list of complexnumbersΛ = {λ1, λ2, . . . , λn} is the spectrumof ann × nentrywise

nonnegative matrix, where λ1 is the Perron eigenvalue and λ2 ∈ R, then for any t � 0 the list

Λt = {λ1 + t, λ2 ± t, λ3, . . . , λn}is also the spectrum of an entrywise nonnegative matrix.

In [2], the authors showed that if Λ = {λ1, λ2, . . . , λn}, with λ1 > λ2 � · · · � λn � 0, is the spec-

trum of a positive matrix A ∈ CSλ1, then Λt = {λ1 + t, λ2 ± t, λ3, . . . , λn} is also the spectrum of a

positivematrix B ∈ CSλ1+t . Alternative results to Theorem1.2were also showed in [2] for positive gen-

eralized doubly stochastic matrices with nonnegative spectrum. Under certain constraints [2, Lemmas

2.2 and 2.7], the results still hold if Λ contains some negative numbers. The arguments used to prove

the results in [2] do not permit to relax the hypotheses to nonnegativematrices. They need of a positive

initial matrix A.

In this paper we extend the result of Guo to elementary divisors. Besides, unlike [2], our ar-

guments allow us to consider nonnegative matrices. In particular, if A is an nonnegative general-

ized stochastic matrix with spectrum Λ = {λ1, λ2, . . . , λn} and elementary divisors (λ − λ1), (λ −λ2)

n2 , . . . , (λ − λk)nk ; n2 + · · · + nk = n − 1, we construct, by means of two rank one perturba-

tions, a modified matrix B, which is also nonnegative generalized stochastic, with spectrum Λt ={λ1 + t, λ2 ± t, λ3, . . . , λn} and elementary divisors (λ − λ1 − t), (λ − λ2 ∓ t), (λ − λ2)

n2−1, (λ −λ3)

n3 , . . . , (λ − λk)nk ; n2 + · · · + nk = n − 1.

We recall the following result given in [6]:

Lemma 1.1 [6]. Let q = (q1, . . . , qn)T be an arbitrary n-dimensional vector and let A ∈ CSλ1

with JCF

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548 J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555

S−1AS = J(A) =

⎡⎢⎢⎢⎣

λ1

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎦ .

Letλ1 + ∑ni=1 qi /= λi, i = 2, . . . , n. Then thematrix A + eqT has JCF J(A) + (∑n

i=1 qi)E11. In particular,

if∑n

i=1 qi = 0 then A and A + eqT are similar.

The paper is organized as follows: In Section 2 we introduce some preliminary results, which will

be used in the next section. In Section 3 we consider perturbation results, which allow us to construct,

from a nonnegative A ∈ CSλ1, a modified matrix B ∈ CSλ1+t with the desired spectral properties. We

also show, in section 4, some examples to illustrate the results.

2. Preliminary results

We start this section with the following result, which will be useful to prove the main results:

Theorem 2.1. Let the n × n matrices

A =[a xT

0 Q

]and B =

[a 0T

0 Q

],

where a ∈ C, x ∈ Cn−1, 0 = (0, . . . , 0)T and Q is an (n − 1) × (n − 1) complex matrix. If a is not an

eigenvalue of Q , then the matrices A and B are similar.

Proof. Let

S =[1 0T

e In−1

], xT = [x1, . . . , xn−1] .

Then

A = B + e1 [0, x1, . . . , xn−1]

and

SAS−1 = SBS−1 + Se1 [0, x1, . . . , xn−1] S−1

= SBS−1 + e

⎡⎣−

n−1∑i=1

xi, x1, x2, . . . , xn−1

⎤⎦ .

Observe that SBS−1, SAS−1 ∈ CSa and

J(SBS−1) =[a

J(Q)

]=

⎡⎢⎢⎢⎣

a

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎦ ,

with a /= λi, i = 2, . . . , k. Let

qT =⎡⎣−

n−1∑i=1

xi, x1, . . . , xn−1

⎤⎦ .

Then qTe = 0 and from Lemma 1.1, the matrices SBS−1, SBS−1 + eqT = SAS−1 are similar. Hence, A

and B are also similar. �

The following result is an extension of Lemma 1.1 (Lemma 3.1 in [6])

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J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555 549

Lemma 2.1. Let A be an n × n complex matrix with Av = λ1v, v /= 0. Let

J(A) =

⎡⎢⎢⎢⎣

Jn1(λ1)Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎦ ; n1 + · · · + nk = n,

be the JCF of A. Let q = (q1, . . . , qn)T ∈ Cn such that λ1 + qTv is a simple eigenvalue of A + vqT . Then

J(A + vqT ) =

⎡⎢⎢⎢⎢⎢⎢⎣

λ1 + qTv

Jn1−1(λ1)Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

In particular, if n1 = 1, then J(A + vqT ) = J(A) + qTvE11. If ma(λ1) = 1 and qTv = 0 then A and

A + vqT are similar.

Proof. Let S = [v|S(2)| · · · |S(n)] be an n × n nonsingular matrix such that A = SJ(A)S−1. Then

S−1(A + vqT )S = J(A) + S−1vqT S

= J(A) + e1qT S

=

⎡⎢⎢⎢⎢⎢⎢⎣

λ1 + qTv ∗ ∗ · · · ∗Jn1−1(λ1)

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

. (1)

Hence, from Theorem 2.1,

J(A + vqT ) =

⎡⎢⎢⎢⎢⎢⎢⎣

λ1 + qTv

Jn1−1(λ1)Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

.

Moreover, if n1 = 1

J(A + vqT ) =

⎡⎢⎢⎢⎢⎣

λ1 + qTv

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎦

= J(A) + qTvE11.

If qTv = 0 andma(λ1) = 1 then from (1) and Theorem 2.1, J(A + vqT ) = J(A). �

Theorem 2.2. Let Λ = {λ1, λ2, . . . , λn} be the spectrum of an n × n nonnegative matrix A,with ρ(A) =λ1 and elementary divisors

(λ − λ1)n1 , (λ − λ2)

n2 , . . . , (λ − λk)nk ; n1 + · · · + nk = n.

Let t > 0. Then, Λt = {λ1 + t, λ2, . . . , λn} is also the spectrum of an n × n nonnegative matrix B with

elementary divisors

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550 J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555

(λ − λ1 − t), (λ − λ1)n1−1, (λ − λ2)

n2 , . . . , (λ − λk)nk ; n1 + · · · + nk = n.

Proof. Let

J(A) =

⎡⎢⎢⎢⎣

Jn1(λ1)Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎦

be the JCF of A. Let Av = λ1v, with v = (v1, . . . , vn)T ∈ Rn, v /= 0, v � 0. Let t > 0. There exist a real

nonnegative vector qT = (q1, q2, . . . , qn), such that t = qTv. Then, A + vqT � 0 and from Theorem 1.1,

σ(A + vqT ) = {λ1 + t, λ2, . . . , λn} = Λt .

From Lemma 2.1,

J(A + vqT ) =

⎡⎢⎢⎢⎢⎢⎢⎣

λ1 + t

Jn1−1(λ1)Jn2(λ2)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

.

Therefore, for t > 0, Λt is the spectrum of a nonnegative matrix B = A + vqT with ρ(B) = λ1 + t

and elementary divisors

(λ − λ1 − t), (λ − λ1)n1−1, (λ − λ2)

n2 , . . . , (λ − λk)nk . �

Theorem 2.3. Let Λ = {λ1, λ2, . . . , λn} be the spectrum of an n × n nonnegative generalized doubly

stochastic matrix A, ρ(A) = λ1 and elementary divisors

(λ − λ1), (λ − λ2)n2 , . . . , (λ − λk)

nk ; n2 + · · · + nk = n − 1.

Let t > 0. Then, Λt = {λ1 + t, λ2, . . . , λn} is also the spectrum of a nonnegative generalized doubly

stochastic matrix B, ρ(B) = λ1 + t and elementary divisors

(λ − λ1 − t), (λ − λ2)n2 , . . . , (λ − λk)

nk ; n2 + · · · + nk = n − 1.

Proof. Let

J(A) =

⎡⎢⎢⎢⎣

λ1

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎦

be the JCF of A. Let q =(tn, tn, . . . , t

n

)T. Then eqT � 0 and qTe = t. Hence

A + eqT � 0,

σ(A + eqT ) = {λ1 + t, λ2, . . . , λn} = Λt ,

with

(A + eqT )e = Ae + eqTe

= λ1e + (qTe)e

= λ1e + te

= (λ1 + t)e

and

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J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555 551

(A + eqT )Te = ATe + qeTe

= λ1e + nq

= λ1e + te

= (λ1 + t)e.

From Lemma 1.1, J(A + eqT ) = J(A) + tE11. Therefore, there exists B = A + eqT � 0with the required

spectral properties. �

Corollary 2.1. Let Λ = {λ1, λ2, . . . , λn} be the spectrum of an n × n symmetric nonnegative generalized

doubly stochastic matrix A, ρ(A) = λ1. Let t > 0. Then, Λt = {λ1 + t, λ2, . . . , λn} is also the spectrum

of a symmetric nonnegative generalized doubly stochastic matrix B, with ρ(B) = λ1 + t.

Proof. From Theorem 2.3 it only remains to show that B = A + eqT is symmetric. For this, observe

that eqT = qeT . Then

BT = (A + eqT )T

= AT + qeT

= A + eqT

= B. �

3. Perturbation results

In this section we introduce the main results, which extend Theorem 1.2 to elementary divisors.

Theorem 3.1. Let Λ = {λ1, λ2, . . . , λn} be the spectrum of an n × n nonnegative generalized stochastic

matrix A, with λ1 > λ2 � · · · � λp real, 2� p� n, ρ(A) = λ1 and elementary divisors

(λ − λ1), (λ − λ2)n2 , . . . , (λ − λk)

nk ; n2 + · · · + nk = n − 1.

Let t > 0. Then, Λt = {λ1 + t, λ2 + t, λ3, . . . , λn} is also the spectrum of a nonnegative generalized

stochastic matrix B, ρ(B) = (λ1 + t), and elementary divisors

(λ − λ1 − t), (λ − λ2 − t), (λ − λ2)n2−1, (λ − λ3)

n3 , . . . , (λ − λk)nk ;

with n2 + · · · + nk = n − 1.

Proof. Although it is clear, from Theorem 1.2 and the result of Johnson [4], that Λt is the spectrum of

a nonnegative generalized stochastic matrix, we here include, for the sake of completeness, this part

of the proof. Let v = (v1, v2, . . . , vn)T ∈ Rn, v /= 0, and Av = λ2v. Let

A = S

⎡⎢⎢⎢⎣

λ1

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎦ S−1,

where S = [e|v|S3| . . . |Sn]. LetvM = max {v1, v2, . . . , vn} ,vm = min {v1, v2, . . . , vn} ,

withM,m ∈ {1, . . . , n}. Let B = A + eqTe + vqT

v , where

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552 J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555

qe = − tvm

vM − vmeM + tvM

vM − vmem

and

qv = t

vM − vmeM − t

vM − vmem.

Then eqTe + vqT

v � 0 and B is nonnegative. Moreover since

qTev = qT

ve = 0, qTee = qT

vv = t,

then

Be =(A + eqT

e + vqTv

)e

= Ae + eqTee + vqT

ve

= Ae + te

= (λ1 + t)e

and

(A + eqTe)v = Av + e

(qTev

)= Av = λ2v.

Hence,

σ(A + eqTe) = {λ1 + t, λ2, . . . , λn} and

σ((

A + eqTe

)+ vqT

v

)= {λ1 + t, λ2 + t, λ3, . . . , λn} = Λt

with λ2 + t = λ2 + qTvv simple eigenvalue of B. Now, from Lemma 2.1

J(A + eqTe) =

⎡⎢⎢⎢⎣

λ1 + t

Jn2(λ2). . .

Jnk(λk)

⎤⎥⎥⎥⎦ .

Let P a permutation matrix such that

PJ(A + eqTe)P

−1 =

⎡⎢⎢⎢⎢⎢⎢⎣

Jn2(λ2)λ1 + t

Jn3(λ3). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

.

Then from Lemma 2.1

J((

A + eqTe

)+ vqT

v

)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ2 + t

Jn2−1(λ2)λ1 + t

Jn3(λ3). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

and the result follows. �

Now we consider a list Λ = {λ1, λ2, . . . , λn} ⊂ C, with p real numbers, p� n, in the following

cases:

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J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555 553

(a) ma(λ1) = p or

(b) ma(λ1) = 1, ma(λ2) = p − 1

}(2)

Theorem 3.2. Let Λ = {λ1, λ2, . . . , λn}, as in (2), be the spectrum of an n × n nonnegative generalized

stochastic matrix A, ρ(A) = λ1 and elementary divisors

(λ − λ1), (λ − λ2)n2 , . . . , (λ − λk)

nk ; n2 + · · · + nk = n − 1.

Let t > 0. Then, Λt = {λ1 + t, λ2 − t, λ3, . . . , λn} is also the spectrum of a nonnegative generalized

stochastic matrix B, ρ(B) = (λ1 + t), and elementary divisors

(λ − λ1 − t)), (λ − λ2 + t)), (λ − λ2)n2−1, (λ − λ3)

n3 , . . . , (λ − λk)nk ;

n2 + · · · + nk = n − 1.

Proof. Similarly to the proof of Theorem 3.1, let (B = A + eqTe + vqT

v , where

qe = tvM

vM − vmeM − tvm

vM − vmem

and

qv = − t

vM − vmeM + t

vM − vmem.

Then B is nonnegative and Be = (λ1 + t)e. Since

B = A + eqTe + vqT

v

= SJ(A)S−1 + Se1

(qTeS

)S−1 + Se2

(qTvS

)S−1

= S[J(A) + e1

(qTeS

)+ e2

(qTvS

)]S−1,

where qTeS = (t, ∗, . . . , ∗) and qT

vS = (0,−t, ∗, . . . , ∗) and

J(A) + e1

(qTeS

)+ e2

(qTvS

)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1 + t ∗ ∗ ∗ · · · ∗λ2 − t ∗ ∗ · · · ∗

Jn2−1(λ2)Jn3(λ3)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,

then

σ(B) = {λ1 + t, λ2 − t, λ3, . . . , λn} = Λt ,

with λ1 + t, λ2 − t simple eigenvalues.

From Theorem 2.1

J(A) + e1

(qTeS

)+ e2

(qTvS

)is similar to⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1 + t

λ2 − t ∗ ∗ · · · ∗Jn2−1(λ2)

Jn3(λ3). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

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554 J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555

and the matrices⎡⎢⎢⎢⎢⎢⎢⎣

λ2 − t ∗ ∗ · · · ∗Jn2−1(λ2)

Jn3(λ3). . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

and ⎡⎢⎢⎢⎢⎢⎢⎣

λ2 − t

Jn2−1(λ2)Jn3(λ3)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎦

are also similar. Then

J(A) + e1

(qTeS

)+ e2

(qTvS

)is also similar to⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1 + t

λ2 − t

Jn2−1(λ2)Jn3(λ3)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

and

J(B) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

λ1 + t

λ2 − t

Jn2−1(λ2)Jn3(λ3)

. . .

Jnk(λk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Hence B has elementary divisors

(λ − λ1 − t), (λ − λ2 + t)), (λ − λ2)n2−1, (λ − λ3)

n3 , . . . , (λ − λk)nk ;

n2 + · · · + nk = n − 1. �

4. Examples

Example 4.1. In [5, Theorem 8] it was shown that the list Λ = {6, 1, 1,−4,−4} is realizable as the

spectrum of the nonnegative matrix

A =

⎡⎢⎢⎢⎢⎢⎣

0 4 0 0 2

4 0 0 0 232

0 0 4 12

32

0 4 0 12

0 0 6 0 0

⎤⎥⎥⎥⎥⎥⎦,

which has elementary divisors (λ − 6), (λ − 1)2, (λ + 4), (λ + 4). By applying Theorem 3.1 we con-

struct, for any t � 0, a nonnegative matrix B ∈ CS6+t ,

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J. Ccapa, R.L. Soto / Linear Algebra and its Applications 432 (2010) 546–555 555

B =

⎡⎢⎢⎢⎢⎢⎣

t 4 0 0 2

t + 4 0 0 0 212t + 3

20 0 4 1

2t + 1

212t + 3

20 4 0 1

2t + 1

20 0 6 0 t

⎤⎥⎥⎥⎥⎥⎦,

with Jordan canonical form J(B) = diag{6 + t, 1 + t, 1,−4,−4}.Example 4.2. In [6] it was shown that

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 2 2 1 1 1 1

2 0 3 1 1 1 1

3 2 0 1 1 1 1

3 2 2 0 0 1 1

0 2 2 2 0 2 1

3 2 2 1 1 0 0

1 2 2 1 1 2 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

has elementary divisors (λ − 9), (λ + 2)2, (λ + 1 − i)2, (λ + 1 + i)2. By applying Theorem 3.2 we

construct, for t > 0, the nonnegative matrix

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 t + 2 2 1 1 1 1

t + 2 0 3 1 1 1 1

3 t + 2 0 1 1 1 1

3 t + 2 2 0 0 1 1

0 t + 2 2 2 0 2 1

3 t + 2 2 1 1 0 0

1 t + 2 2 1 1 2 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

with elementary divisors (λ − 9 − t), (λ + 2 + t), (λ + 2), (λ + 1 − i)2, (λ + 1 + i)2.

References

[1] A. Brauer, Limits for the characteristic roots of amatrix IV: applications to stochasticmatrices,DukeMath. J. 19 (1952) 75–91.[2] J. Ccapa, R.L. Soto, On spectra perturbation and elementary divisors of positivematrices, Electron. J. Linear Algebra 18 (2009)

462–481.[3] W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270.[4] C.R. Johnson, Row stochastic matrices similar to doubly stochastic matrices, Linear Multilinear Algebra 10 (1981) 113–130.[5] R.L. Soto, O. Rojo, Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl.

416 (2006) 844–856.[6] R.L. Soto, J. Ccapa, Nonnegativematriceswith prescribed elementary divisors, Electron. J. Linear Algebra 17 (2008) 287–303.